Let $F = \mathbb{C}$
Let V be a finite-dimensional complex vector space equipped with a positive definite sesquilinear form $\phi$. Let $\alpha$ be a linear operator on $V$, and denote its unique adjoint with respect to $\phi$ by $\alpha^{*}$. Prove the following statements are equivalent:
(a) $\alpha$ is normal
(b)$ \alpha = \alpha_{1}+i\alpha_{2}$ for some self adjoint(with respect to $\phi$ linear operators $\alpha_1$ and $\alpha_2$ on $V$ such that $\alpha_1\circ\alpha_2 = \alpha_2\circ\alpha_1$
(c) $ \alpha^*= g(\alpha)$ for some $g(x) \in \mathbb{C}[x]$
Now for a) ==> c) I have the result that I can use , if $\alpha$ and $\beta$ are linear operators on $V$, $\alpha$ satisfies a polynomial in field $F$, $V$ is $\alpha$-cyclic, then $\alpha$ and $\beta$ commute iff $B= p(a)$ for some polynomial in the field $F$. But afterwards I am just stuck.